When the Tracy-Singh product of matrices represents a certain operation on linear operators

Abstract: Given two linear transformations, with representing matrices $A$ and $B$ with respect to some bases, it is not clear, in general, whether the Tracy-Singh product of the matrices $A$ and $B$ corresponds to a particular operation on the linear transformations. Nevertheless, it is not hard to show that in the particular case that each matrix is a square matrix of order of the form $n^2$, $n>1$, and is partitioned into $n^2$ square blocks of order $n$, then their Tracy-Singh product, $A \boxtimes B$, is similar to $A \otimes B$, and the change of basis matrix is a permutation matrix. In this note, we prove that in the special case of linear operators induced from set-theoretic solutions of the Yang-Baxter equation, the Tracy-Singh product of their representing matrices is the representing matrix of the linear operator obtained from the direct product of the set-theoretic solutions.